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# **Capacitated Vehicle Routing Problem（CVRP）详解与Python代码示例**

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# **一、问题详解**

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# Capacitated Vehicle Routing Problem（CVRP）是一个在物流管理和运筹学中广泛研究的问题。它涉及到如何为一组已知需求的客户设计从中心仓库出发并返回的最小费用路径，同时需要满足一系列约束条件，如每个客户只能被服务一次，车辆的载重能力有限等。

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# 在CVRP中，我们假设有一个中心仓库（或称为配送中心），以及多个需要配送货物的客户点。每个客户点都有一定数量的货物需求，而车队中的每辆车都有一定的载重能力。我们的目标是设计一条或多条从仓库出发，经过所有客户点并返回仓库的路径，使得在满足所有约束条件的前提下，总运输成本最低。

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# **二、Python代码示例**

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# 以下是一个使用Google的OR-Tools库解决CVRP问题的Python代码示例。OR-Tools是一个开源的优化软件工具包，其中包含了许多用于解决各种优化问题的算法和工具。

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from ortools.constraint_solver import routing_enums_pb2

from ortools.constraint_solver import pywrapcp



def create_data_model(num_locations, num_vehicles, depot, demands, capacity):

    """创建数据模型"""

    data = {}

    data['num_locations'] = num_locations

    data['num_vehicles'] = num_vehicles

    data['depot'] = depot

    data['demands'] = demands

    data['vehicle_capacities'] = [capacity] * num_vehicles

    data['distance_matrix'] = [...]  # 这里应填入距离矩阵，例如使用Floyd-Warshall算法计算

    return data



def print_solution(data, manager, routing, assignment):

    """打印解决方案"""

    total_distance = 0

    for vehicle_id in range(data['num_vehicles']):

        index = routing.Start(vehicle_id)

        plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)

        while not routing.IsEnd(index):

            plan_output += ' {} -> '.format(manager.IndexToNode(index))

            total_distance += routing.GetArcCostForVehicle(index, index + 1, vehicle_id)

            index = assignment.Value(routing.NextVar(index))

        plan_output += '{}\n'.format(manager.IndexToNode(index))

        plan_output += 'Distance of the route: {}m\n'.format(total_distance)

        print(plan_output)



def main():

    """主函数"""

    # 示例数据

    num_locations = 10  # 地点数量，包括仓库和客户点

    num_vehicles = 3  # 车辆数量

    depot = 0  # 仓库索引

    demands = [0, 10, 20, 15, 10, 20, 25, 20, 30, 10]  # 客户点需求

    capacity = 50  # 车辆容量



    # 创建数据模型

    data = create_data_model(num_locations, num_vehicles, depot, demands, capacity)



    # 创建路由模型

    routing = pywrapcp.CreateRoutingIndexManager(data['num_locations'], data['num_vehicles'], data['depot'])



    # 创建路由搜索参数

    search_parameters = pywrapcp.DefaultRoutingSearchParameters()



    # 定义并添加容量约束

    routing_model = pywrapcp.RoutingModel(routing, search_parameters)

    capacity_dimension = routing_model.AddDimension(

        data['demands'], 0, data['vehicle_capacities'],

        True, 'Capacity')



    # 定义并设置成本（这里假设成本就是距离）

    for from_index in range(data['num_locations']):

        for to_index in range(data['num_locations']):

            if from_index != to_index:

                routing_model.AddArcWithCost(from_index, to_index, data['distance_matrix'][from_index][to_index])



    # 求解问题

    assignment = routing_model.SolveWithParameters(search_parameters)



    # 打印解决方案

    print_solution(data, routing, routing_model, assignment)



if __name__ == '__main__':

    main()
